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Chapter 2: Problem 31

Find all real solutions of the polynomial equation. $$2 y^{4}+3 y^{3}-16 y^{2}+15 y-4=0$$

Short Answer

Expert verified
The valid real solutions for the given polynomial equation are \( y = -1,1/2,-2,-1/2 \)

Step by step solution

01

Write the Polynomial Equation

First, write down the given polynomial equation, which is \(2y^4 + 3y^3 - 16y^2 + 15y - 4 = 0\). The goal is to factorize this polynomial to make it tractable.
02

Factorize Polynomial to Unveil Quadratic Form

The equation can be rewritten by factoring it as \( (2y^2 - y - 2)(y^2 + 2y - 2) = 0 \). This is achieved by factorizing the given polynomial until it can be expressed in the form of two quadratic equations.
03

Solve Quadratic Equations

The two quadratic equations are now \(2y^2 - y - 2 = 0\) and \(y^2 + 2y - 2 = 0\). Solve for \(y\) in each equation using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Solving each quadratic equation will give two solutions for each equation, resulting in a total of four possible solutions for \(y\).
04

Testing Solutions

Lastly, some solutions might be complex roots, which are not considered real, therefore they must be discarded. Hence, only the real roots should be considered as valid solutions.

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Most popular questions from this chapter

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$g(x)=x^{5}-8 x^{4}+28 x^{3}-56 x^{2}+64 x-32$$

For each function, identify the degree of the function and whether the degree of the function is even or odd. Identify the leading coefficient and whether the leading coefficient is positive or negative. Use a graphing utility to graph each function. Describe the relationship between the degree of the function and the sign of the leading coefficient of the function and the right-hand and left-hand behavior of the graph of the function. (a) \(f(x)=x^{3}-2 x^{2}-x+1\) (b) \(f(x)=2 x^{5}+2 x^{2}-5 x+1\) (c) \(f(x)=-2 x^{5}-x^{2}+5 x+3\) (d) \(f(x)=-x^{3}+5 x-2\) (e) \(f(x)=2 x^{2}+3 x-4\) (f) \(f(x)=x^{4}-3 x^{2}+2 x-1\) (g) \(f(x)=x^{2}+3 x+2\)

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=2 x^{4}+5 x^{3}+4 x^{2}+5 x+2$$

Use the position equation $$s=-16 t^{2}+v_{0} t+s_{0}$$ where \(s\) represents the height of an object (in feet), \(v_{0}\) represents the initial velocity of the object (in feet per second), \(s_{0}\) represents the initial height of the object (in feet), and \(t\) represents the time (in seconds). A projectile is fired straight upward from ground level \(\left(s_{0}=0\right)\) with an initial velocity of 160 feet per second. (a) At what instant will it be back at ground level? (b) When will the height exceed 384 feet?

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=x^{3}-4 x^{2}+1\) (a) Upper: \(x=4\) (b) Lower: \(x=-1\)
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